Question

数列{an}中,a1=1,当n≥2时,a1a2a3···an=n²,比较an,a（n+1）的大小并说明理由

Answer 1

a1*a2*a3*……*an= n²

a1*a2*a3*……*a(n-1)= (n-1)²

将第一式除第二式,得:an=n²/(n-1)²,an>0

同理：a(n+1)=(n+1)²/n²

那么：a(n+1)/an = [(n+1)²/n²] / [n²/(n-1)²]

= (n+1)²(n-1)²/ (n²)²

= [(n²-1)/n²]²

= (1- 1/n²)² a(n+1)